We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring to be equal to its inner rank . We apply it in two different contexts.
In our first application, we reprove a recent result of T. Mai, R. Speicher and S. Yin: if are operators in a finite von Neumann algebra with a faithful normal trace , then they generate the free division ring on in the algebra of unbounded operators affiliated to if and only if the space of tuples of finite rank operators on satisfying
In our second and main application we construct explicitly the universal division ring of fractions of , where is a division ring, and we use it in order to show the following instance of pro- Lück approximation.
Let be a finitely generated free pro -group, a chain of normal open subgroups of with trivial intersection and a matrix over . Denote by the matrix over obtained from the matrix by applying the natural homomorphism . Then there exists the limit
and it is equal to the inner rank of the matrix .
Cite this article
Andrei Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of . J. Comb. Algebra 4 (2020), no. 4, pp. 369–395DOI 10.4171/JCA/47